# AP Statistics Curriculum 2007 IntroDesign

General Advance-Placement (AP) Statistics Curriculum - Design and Experiments

## Design of Experiments

Design of experiments refers of the blueprint for planning a study or experiment, performing the data collection protocol and controlling the study parameters for accuracy and consistency. Design of experiments only makes sense in studies where variation, chance and uncertainly are present and unavoidable. Data, or information, is typically collected in regard to a specific process or phenomenon being studied to investigate the effects of some controlled variables (independent variables or predictors) on other observed measurements (responses or dependent variables). Both types of variables are associated with specific observational units (living beings, components, objects, materials, etc.)

## Approach

The following are the most common components used in Experimental Design:

### Comparison

To make inference about effects, associations or predictions, one typically has to compare different groups subjected to distinct conditions. This allows contrasting observed responses and underlying group differences that may ultimately lead to inference on relationships and influences between controlled and observed variables.

### Randomization

The second fundamental design principle is randomization. It requires that we make allocation of (controlled variables) treatments to units using some random mechanism. This simply guarantees that the effects resulted are the units, but not incorporated in the model, equidistributed amongst all groups and therefore unlikely to significantly affect our group comparisons at the end of the statistical inference or analysis (as these effects, if present, will be similar within each group).

### Experimental vs. Observational Studies

There are many situations, where randomized experiments, albeit desirable, are impractical. And therefore, we can not deduce causality or effects of various treatments on the response measurement. Observational studies are retrospective or prospective studies where the investigator does not have control over randomization of treatments to subject, or units. In these cases, the subjects or units fall naturally within a treatment group. Examples of such observational studies include studies of smoking effects on cancer and use of parachutes to prevent death and major trauma related to gravitational pull.

### Replication

All measurements, observations or data collected are subject to variation, as there are no completely deterministic processes. As we try to make inference about the process that generated the observed data (not the sample data itself, even though our statistical analysis are data-driven and based on the observed measurements), the more data we collect (unbiasly) the stronger our inference is likely to be. Therefore, repeated measurements intuitively would allow is to tame the variability associated with the phenomenon we study.

### Blocking

Blocking is related to randomization. The difference is that we use blocking when we know a priori of certain effects of the observational units on the response measurements (e.g., when studying the effects of hormonal treatments on humans, gender plays a significant role). We arrange units into groups (blocks) that are similar to one another when we design an experiment in which certain unit characteristics are known to affect the response measurements. Blocking reduces known and irrelevant sources of variation between units and allows greater precision in the estimation of the source of variation in the study.

### Orthogonality

Orthogonality allows division of complex relations, variation into separate (independent/orthogonal) contrasts, or factors, that can be studied efficiently and autonomously. Often, these contrasts may be represented by vectors, where sets of orthogonal contrasts are uncorrelated and may be independently distributed. Independence implies that each orthogonal contrast provides complementary information to other contrasts (i.e., other treatments). The goal is to completely decompose the variance or the relations of the observed measurements into independent components (e.g., like Taylor expansion allowing polynomial decomposition of smooth functions, where the polynomial base functions are easy to differentiate, integrate, etc.) This will, of course, allow easier interpretation of the statistical analysis and the findings of the study.

## Model Validation

All of the components in the approach/methods section need to be validated but the major one is the independence assumption.

## Examples & hands-on activities

### A study of aortic valve-sparing repair*

• This study sought to establish whether there was a difference in outcome after aortic valve repair with autologous pericardial leaflet extension in acquired versus congenital valvular disease. In this study, 128 patients underwent reparative aortic valve surgery at UCLA from 1997 through 2005 for acquired or congenital aortic valve disease. The acquired group (43/128) (34%) had a mean age of 56.4 $\pm$ 20.3 years (range, 7.8—84.6 years) and the congenital group (85/128) (66%) had a mean age of 16.9 $\pm$19.2 years (range, 0.3—82 years). The endpoints of the study were mortality and reoperation rates.
• In this case the units are heart disease patients. These were split into two groups (acquired or congenital) and blocked by gender (male/female). The treatment allied on the two groups was aortic valve repair with autologous pericardial leaflet extension.